I am learning about multigrid methods from the textbook section 6.3 Multigrid Methods, which shows a geometric multigrid algorithm for 1-D examples in detail, including how to build restriction/interpolation matrices (i.e., eqs.1, 3) and how to get a coarse coefficient matrix (eq. 6).

I realize that these matrices and way of coarsening the coefficient matrix are important for multigrid methods.

But for the 2-D case, only the ways of getting restriction and interpolation matrices are shown in the book, not the coefficient matrix for coarse grid (i.e., $A_{2h}$, $A_{4h}$, etc.).

This missing link is barrier for me to go further to learn multigird methods.

Can anybody help me?

  • $\begingroup$ Gilbert Strang is a known name in many fields and to me, this document is intended to be just a starter document - more like a lecture note. Volker John also has a good set of lecture notes that you might be interested in: wias-berlin.de/people/john/LEHRE/MULTIGRID/multigrid.pdf . However, in my opinion, you should ask for references to learn about multigrid methods. I am pretty sure there are many great books out there about it. $\endgroup$ – Abdullah Ali Sivas Jul 19 '20 at 16:55

For geometric multigrid, the $\mathbf A_{2h}$ and $\mathbf A_{4h}$ (etc) matrices are just discretizations of the same PDE on coarser grids.

For instance, if your original $\mathbf A_{h}$ was a finite difference approximation of the laplacian on a 64 point grid, then $\mathbf A_{2h}$ would be a finite difference approximation of the laplacian on a 32 point grid, etc. You should also see some sort of relationship where $\mathbf A_{2h} = \mathbf R \mathbf A_h \mathbf R^T$ (maybe to within a constant? I forget).

In algebraic multigrid, only the latter relationship really applies. There is no "fine grid" or "coarse grid", just the matrices themselves, and the restriction operator itself uniquely defines the coarsened problem(s).

  • $\begingroup$ Actually, all this answer would seem to already be contained in your reference, see Strang's equation (6), combined with the fact that restriction (R) and interpolation (I) are transposes of each other, as Strang remarks just above his equation (3). $\endgroup$ – rchilton1980 Jul 20 '20 at 15:31
  • $\begingroup$ But I have no idea about what the matrix R is for 2-D and 3-D problems $\endgroup$ – Freewill Jul 28 '20 at 16:19
  • $\begingroup$ These matrices just represent interpolation-like procedures, the 2D/3D version of R' is just comes from the tensor product of the 1D version. (That is, the 1D version of R is linear interpolation, the 2D/3D versions of R are just bilinear and trilinear interpolation). $\endgroup$ – rchilton1980 Aug 19 '20 at 16:08

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